The invention relates generally to quantum mechanical signal amplifiers. In particular, the invention relates to a triple Mach-Zehnder interferometer (MZI) configuration for augmenting signals by determining photon intensity via weak measurements.
I. Overview:
The weak value Aw of a quantum mechanical observable Â was introduced by Aharonov et al. a quarter century ago. See Y. Aharonov et al., Ann. N.Y. Acad. Sci. 480 417-421 (1986); Y. Aharonov et al., Phys. Rev. Lett. 60 1351 (1988); and Y. Aharonov et al., Phys. Rev. A 41 11 (1990). This quantity is the statistical result of a standard measurement procedure performed upon a pre- and post-selected (PPS) ensemble of quantum systems when the interaction between the measurement apparatus and each system is sufficiently weak, i.e., when it is a weak measurement. Unlike a standard strong measurement of observable Â, which significantly disturbs the measured system (i.e., such interrogation “collapses” the wave function), a weak measurement of Â for a PPS system does not appreciably disturb the quantum system and yields Aw as the observable's measured value.
The peculiar nature of the virtually undisturbed quantum reality that exists between the boundaries defined by the PPS states is revealed in the eccentric characteristics of Aw namely that Aw is complex valued and that the real Re Aw and imaginary Im Aw of Aw can be extremely large and lie far outside the eigenvalue spectral limits of Â. This causes the pointer of a measurement apparatus to experience translations much greater than those obtained from standard strong measurements.
Although the interpretation of weak values remains somewhat controversial, experiments have verified several of the interesting unusual properties predicted by weak value theory. See N. Richie et al., Phys. Rev. Lett. 66 1107 (1991); A. Parks et al., Proc. R. Soc. A 454 2997 (1998); K. Resch et al., Phys. Lett. A 324 125 (2004); Q. Wang et al., Phys. Rev A 73 023814 (2006); O. Hosten et al., Science 319 787 (2008); Y. Yokota et al., New J. Phys. 11 033011 (2009); and P. Dixon et al., Phys. Rev. Lett. 102 173601 (2009).
II. Weak Values:
Weak measurements arise in the von Neumann description of a quantum measurement at time t0 of a time independent observable Â that describes a quantum system in an initial pre-selected state |ψ1>=Σjcj|aj> at t0, where the set J indexes the eigenstates |aj> of observable Â. In this description, the Hamiltonian for the interaction between the measurement apparatus and the quantum system is:Ĥ=γ(t)Â{circumflex over (p)},  (1)where γ(t)=γδ(t−t0) defines the strength of the measurement's impulsive coupling interaction at t0 and {circumflex over (p)} is the momentum operator for the pointer of the measurement apparatus which is in the initial normalized state |φ>.
Let {circumflex over (q)} be the pointer's position operator that is conjugate to {circumflex over (p)}. Prior to the measurement, the pre-selected system and the pointer are in the tensor product state |ψ1>|φ>. Immediately following the interaction, the combined system is in the state:
                                                                                                    ⁢                                                                  Φ                  〉                                =                                ⁢                                  ⁢                                      ⅇ                                                                  -                                                  ⅈ                          ℏ                                                                    ⁢                                              ∫                                                                              H                            .                                                    ⁢                                                      ⅆ                            t                                                                                                                                ⁢                                                                                ψ                      i                                        〉                                    ⁢                                                          φ                    〉                                                                                                                                          =                                ⁢                                  ⁢                                                                                                        ⅇ                                                                              -                                                          ⅈ                              ℏ                                                                                ⁢                          γ                          ⁢                                                      A                            .                                                    ⁢                                                      p                            .                                                                                              ⁢                                              ψ                        i                                                              〉                                    ⁢                                                          φ                    〉                                                              ,                                                          (        2        )            where
          ⁢      ⁢          ⅇ                        -                      ⅈ            ℏ                          ⁢        γ        ⁢                  A          .                ⁢                  p          .                    is the von Neumann measurement operator. If the state |ψf>=ΣJc′j|aj> <ψf|ψl>≠0 is post-selected at t0, then the resulting pointer state is:|Ψ>≡<ψf|Φ>=<ψf|e−iγÂ{circumflex over (p)}/n|ψl>|φ>.  (3)
A weak measurement of observable Â occurs when the interaction strength γ is sufficiently small so that the system is essentially undisturbed, and the pointer's uncertainty Δq is much larger than Â's eigenvalue separation. In this case, eqn. (3) becomes:
                                                      Ψ            〉                    ≈                                    〈                                                ψ                  f                                ⁢                                                                                              1                      ^                                        -                                                                  ⅈ                        ℏ                                            ⁢                      γ                      ⁢                                                                                          ⁢                                              A                        ^                                            ⁢                                              p                        ^                                                                                                              ⁢                                  ψ                  i                                            〉                        ⁢                                        φ              〉                                ≈                                    〈                                                ψ                  f                                |                                  ψ                  i                                            〉                        ⁢                                          S                ^                            ⁡                              (                                  γ                  ⁢                                                                          ⁢                                      A                    w                                                  )                                      ⁢                                        φ              〉                                      ,                                  ⁢        where                            (        4        )                                          A          w                ≡                              〈                                          ψ                f                            ⁢                                                                A                  ^                                                            ⁢                              ψ                i                                      〉                                〈                                          ψ                f                            |                              ψ                i                                      〉                                              (        5        )            is the weak value of observable Â, and the Ŝ(γAw) operator is defined as:
                                              ⁢                                            S              ^                        ⁡                          (                              γ                ⁢                                                                  ⁢                                  A                  w                                            )                                ≡                      ⁢                                          ⅇ                                                      -                                          ⅈ                      ℏ                                                        ⁢                  γ                  ⁢                                                            A                      w                                        .                                    ⁢                                      p                    .                                                              .                                                          (        6        )            
Note that Aw can be calculated from eqn. (5) when the states and Â are known. In order for the measurement to qualify as a weak measurement, the following weakness conditions must be simultaneously satisfied:
                                              ⁢                              Δ            ⁢                                                  ⁢            p            ⁢                          <<                              ℏ                γ                                      ⁢                                                                            A                  w                                                                            -                1                                              ⁢                                          ⁢                                          ⁢          and          ⁢                                          ⁢                                          ⁢          Δ          ⁢                                          ⁢          p          ⁢                      <<                    ⁢                      ℏ            γ                    ⁢                                                                                                          A                    w                                                                              (                                              A                        m                                            )                                        w                                                                                              1                                  m                  -                  1                                                      .                                              (        7        )            This is shown in A. Parks et al., Proc. R. Soc. A 454 2997 (1998) and I. Duck et al., Phys. Rev. D 40 2112 (1989).